Simplify (-34x+4.7)-(3+2.9x): A Step-by-Step Guide
Hey guys! Let's dive into the world of algebra and tackle a question that might seem a bit intimidating at first glance. We're going to break down the expression (-34x + 4.7) - (3 + 2.9x) and figure out what its difference is. Don't worry, it's not as scary as it looks! We'll go through each step together, making sure you understand the logic behind it. By the end of this article, you'll be a pro at simplifying expressions like this one. So, grab your pencils and let's get started!
Understanding the Basics: Variables, Constants, and Expressions
Before we jump into the problem, let's quickly recap some fundamental concepts. In algebra, we deal with variables, which are letters (like 'x' in our case) that represent unknown numbers. Then we have constants, which are just regular numbers like 4.7 and 3. And when we combine variables, constants, and mathematical operations (+, -, *, /), we get an expression. Our expression, (-34x + 4.7) - (3 + 2.9x), is a combination of these elements.
Variables: The Unknowns We Seek
Think of variables as placeholders. They hold the spot for a number we haven't yet determined. The beauty of algebra is that we can manipulate these variables using mathematical rules to eventually solve for their values. In our expression, 'x' is the star of the show – it's the variable we're working with. The term '-34x' means -34 multiplied by the value of 'x'. This coefficient, -34, tells us how many 'x's we have. Understanding this relationship is crucial for simplifying the expression.
Constants: The Steady Numbers
Constants are the numbers that stand alone, without any variables attached. In our expression, we have two constants: 4.7 and 3. These numbers are fixed – they don't change their value based on the value of 'x'. Constants are the anchors in our algebraic seas, providing a stable foundation for our calculations. They're the numbers we can directly add or subtract without worrying about the variable 'x'.
Expressions: Combining Variables and Constants
An expression is a mathematical phrase that combines variables, constants, and operations. It's like a sentence in the language of mathematics. Our expression, (-34x + 4.7) - (3 + 2.9x), is a slightly complex sentence, but we're going to break it down word by word (or term by term) to understand its meaning. The parentheses in our expression indicate that certain operations should be performed together. We'll see how to handle those in the next step.
Step-by-Step Solution: Simplifying the Expression
Now, let's get down to business and simplify the expression (-34x + 4.7) - (3 + 2.9x). We'll take it one step at a time, so you can follow along easily.
Step 1: Distribute the Negative Sign
The first thing we need to do is deal with the parentheses. Notice the minus sign in front of the second set of parentheses: -(3 + 2.9x). This means we need to distribute the negative sign to each term inside the parentheses. It's like multiplying each term by -1. So, the expression becomes:
-34x + 4.7 - 3 - 2.9x
Think of it as if the negative sign is a little ninja sneaking into the parentheses and changing the sign of everything inside. This step is super important because it gets rid of the parentheses and allows us to combine like terms.
Step 2: Combine Like Terms
Now comes the fun part – combining like terms! Like terms are those that have the same variable raised to the same power. In our expression, we have two terms with 'x' (-34x and -2.9x) and two constants (4.7 and -3). We can add or subtract these terms separately.
Let's start with the 'x' terms: -34x - 2.9x. If you think about it, this is like having -34 apples and then losing another 2.9 apples. So, we end up with:
-36.9x
Now, let's combine the constants: 4.7 - 3. This is a simple subtraction:
1.7
Step 3: Write the Simplified Expression
Finally, we put the combined terms together to get our simplified expression:
-36.9x + 1.7
And that's it! We've successfully simplified the expression (-34x + 4.7) - (3 + 2.9x) to -36.9x + 1.7. Pat yourself on the back – you've done it!
Common Mistakes to Avoid
Simplifying expressions is a skill that gets easier with practice, but there are a few common pitfalls you'll want to avoid. Here are some tips to help you stay on track:
Forgetting to Distribute the Negative Sign
This is probably the most common mistake. When you have a minus sign in front of parentheses, remember to distribute it to every term inside. It's easy to forget and only change the sign of the first term, but that will lead to the wrong answer.
Combining Unlike Terms
You can only combine terms that have the same variable raised to the same power. For example, you can combine -34x and -2.9x because they both have 'x' to the power of 1. But you can't combine -34x with 4.7 because 4.7 is a constant, not a term with 'x'.
Making Arithmetic Errors
Simple arithmetic mistakes can throw off your entire solution. Double-check your addition, subtraction, multiplication, and division, especially when dealing with negative numbers. A calculator can be your best friend here, especially for more complex calculations.
Practice Makes Perfect: More Examples
To really master simplifying expressions, it's essential to practice. Let's work through a couple more examples to solidify your understanding.
Example 1: Simplifying (5y - 2) + (3y + 1)
In this example, we have an addition sign between the parentheses, which makes things a little simpler. We don't need to worry about distributing a negative sign. We can simply remove the parentheses and combine like terms.
(5y - 2) + (3y + 1) = 5y - 2 + 3y + 1
Now, let's combine the 'y' terms: 5y + 3y = 8y
And the constants: -2 + 1 = -1
So, the simplified expression is:
8y - 1
Example 2: Simplifying 2(x + 4) - (x - 3)
This example combines distribution with combining like terms. First, we need to distribute the 2 in the first set of parentheses:
2(x + 4) = 2x + 8
Then, we distribute the negative sign in the second set of parentheses:
-(x - 3) = -x + 3
Now, we can rewrite the expression:
2x + 8 - x + 3
Combine the 'x' terms: 2x - x = x
And the constants: 8 + 3 = 11
So, the simplified expression is:
x + 11
Real-World Applications: Why Does This Matter?
You might be wondering, "Okay, this is cool, but when am I ever going to use this in real life?" Well, simplifying expressions is a fundamental skill that has applications in many different fields.
Engineering and Physics
Engineers and physicists use algebraic expressions to model and solve problems in areas like mechanics, electricity, and fluid dynamics. Simplifying these expressions is crucial for making calculations and predictions.
Computer Science
In computer programming, expressions are used to perform calculations and manipulate data. Simplifying expressions can make code more efficient and easier to understand.
Economics and Finance
Economists and financial analysts use algebraic expressions to model economic trends and make investment decisions. Simplifying these expressions can help them identify key relationships and patterns.
Everyday Life
Even in everyday life, you might use simplifying expressions without even realizing it. For example, if you're calculating the total cost of items on sale or figuring out how much paint you need for a room, you're essentially simplifying algebraic expressions.
Conclusion: You've Got This!
Simplifying the expression (-34x + 4.7) - (3 + 2.9x) might have seemed daunting at first, but we've broken it down step by step and shown that it's totally manageable. Remember the key steps: distribute the negative sign, combine like terms, and double-check your work. With practice, you'll become a master of simplifying expressions.
So, keep practicing, stay curious, and don't be afraid to tackle those algebraic challenges. You've got this!
